Let $V$ be a finite dimensional vector space. I'm trying to see if any subspace $A$ of $V^*$ must be an annihilator of some subspace in $U$ of $V$. I.e. I'm trying to see if $A=U^0$ for some subspace $U\subseteq V$.
So first I let $A$ be a subspace of $V^*$. Then I denote the set of all $v\in V$ such that $\phi(v)=0$ for every $\phi\in A$ by $U$.
Lemma: $U$ is a subspace.
Proof: Clearly $0\in U$. Let $u,v \in U$ and $\lambda \in \Bbb F$. Then for every $\phi \in A$ $$\phi(u+\lambda v) = \phi(u) + \lambda\phi(v) = 0 \implies u+\lambda v\in U$$ So $U$ is indeed a subspace of $V$. $\square$
Because $V$ is finite dimensional, so is $V^*$ and thus $A$. Let $\{\phi_1, \dots, \phi_k\}$ be a basis for $A$. Let $u\in U$. Then $\phi_i(u)=0$ by definition. Thus $\phi_i\in U^0$ for all $i=1,\dots k$. So $A\subseteq U^0$.
So the only part that I still need is to show the reverse inclusion or that $\dim A = \dim U^0$. But I can't figure out how to do it.
Here are my problems with the duplicate question (or more specifically its answers):
- user218931 uses the double dual which is introduced in a later exercise.
- Rafael Deiga uses $\dim(\operatorname{null}\phi_1 \cap \cdots \cap \operatorname{null}\phi_m) = \dim V-m$ which is proven in a later exercise.
- Aweygan uses some functional analysis theorem called the Hahn-Banach theorem.
So all of the answers use tools that I don't currently have in my mathematical toolkit. And yes, I realize that there's nothing theoretically wrong with using a result from later in the problem set as long as the proof of that result doesn't rely on this one, but I feel like Axler wouldn't have written out the exercises so that you needed to do that. So to me it feels sorta like cheating to do them out of order. Does anyone see a way to finish this problem without using the above three results?
